Integrand size = 15, antiderivative size = 92 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}+\frac {4 b \left (a+b x^4\right )^{9/4}}{119 a^2 x^{17}}-\frac {32 b^2 \left (a+b x^4\right )^{9/4}}{1547 a^3 x^{13}}+\frac {128 b^3 \left (a+b x^4\right )^{9/4}}{13923 a^4 x^9} \] Output:
-1/21*(b*x^4+a)^(9/4)/a/x^21+4/119*b*(b*x^4+a)^(9/4)/a^2/x^17-32/1547*b^2* (b*x^4+a)^(9/4)/a^3/x^13+128/13923*b^3*(b*x^4+a)^(9/4)/a^4/x^9
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {\left (a+b x^4\right )^{9/4} \left (-663 a^3+468 a^2 b x^4-288 a b^2 x^8+128 b^3 x^{12}\right )}{13923 a^4 x^{21}} \] Input:
Integrate[(a + b*x^4)^(5/4)/x^22,x]
Output:
((a + b*x^4)^(9/4)*(-663*a^3 + 468*a^2*b*x^4 - 288*a*b^2*x^8 + 128*b^3*x^1 2))/(13923*a^4*x^21)
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {803, 803, 803, 796}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \int \frac {\left (b x^4+a\right )^{5/4}}{x^{18}}dx}{7 a}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \int \frac {\left (b x^4+a\right )^{5/4}}{x^{14}}dx}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\right )}{7 a}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \left (-\frac {4 b \int \frac {\left (b x^4+a\right )^{5/4}}{x^{10}}dx}{13 a}-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}\right )}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\right )}{7 a}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {4 b \left (-\frac {8 b \left (\frac {4 b \left (a+b x^4\right )^{9/4}}{117 a^2 x^9}-\frac {\left (a+b x^4\right )^{9/4}}{13 a x^{13}}\right )}{17 a}-\frac {\left (a+b x^4\right )^{9/4}}{17 a x^{17}}\right )}{7 a}-\frac {\left (a+b x^4\right )^{9/4}}{21 a x^{21}}\) |
Input:
Int[(a + b*x^4)^(5/4)/x^22,x]
Output:
-1/21*(a + b*x^4)^(9/4)/(a*x^21) - (4*b*(-1/17*(a + b*x^4)^(9/4)/(a*x^17) - (8*b*(-1/13*(a + b*x^4)^(9/4)/(a*x^13) + (4*b*(a + b*x^4)^(9/4))/(117*a^ 2*x^9)))/(17*a)))/(7*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
orering | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (-128 b^{3} x^{12}+288 a \,b^{2} x^{8}-468 a^{2} b \,x^{4}+663 a^{3}\right )}{13923 x^{21} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
risch | \(-\frac {\left (-128 b^{5} x^{20}+32 a \,b^{4} x^{16}-20 a^{2} b^{3} x^{12}+15 a^{3} b^{2} x^{8}+858 a^{4} b \,x^{4}+663 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{13923 x^{21} a^{4}}\) | \(72\) |
Input:
int((b*x^4+a)^(5/4)/x^22,x,method=_RETURNVERBOSE)
Output:
-1/13923*(b*x^4+a)^(9/4)*(-128*b^3*x^12+288*a*b^2*x^8-468*a^2*b*x^4+663*a^ 3)/x^21/a^4
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {{\left (128 \, b^{5} x^{20} - 32 \, a b^{4} x^{16} + 20 \, a^{2} b^{3} x^{12} - 15 \, a^{3} b^{2} x^{8} - 858 \, a^{4} b x^{4} - 663 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{13923 \, a^{4} x^{21}} \] Input:
integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="fricas")
Output:
1/13923*(128*b^5*x^20 - 32*a*b^4*x^16 + 20*a^2*b^3*x^12 - 15*a^3*b^2*x^8 - 858*a^4*b*x^4 - 663*a^5)*(b*x^4 + a)^(1/4)/(a^4*x^21)
Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (85) = 170\).
Time = 2.49 (sec) , antiderivative size = 954, normalized size of antiderivative = 10.37 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\text {Too large to display} \] Input:
integrate((b*x**4+a)**(5/4)/x**22,x)
Output:
-1989*a**8*b**(37/4)*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x **20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28 *gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) - 8541*a**7*b**(41/4)*x** 4*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256* a**4*b**12*x**32*gamma(-5/4)) - 13734*a**6*b**(45/4)*x**8*(a/(b*x**4) + 1) **(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x** 24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*g amma(-5/4)) - 9786*a**5*b**(49/4)*x**12*(a/(b*x**4) + 1)**(1/4)*gamma(-21/ 4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 7 68*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) - 2625 *a**4*b**(53/4)*x**16*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9* x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**2 8*gamma(-5/4) + 256*a**4*b**12*x**32*gamma(-5/4)) + 231*a**3*b**(57/4)*x** 20*(a/(b*x**4) + 1)**(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x**24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256 *a**4*b**12*x**32*gamma(-5/4)) + 924*a**2*b**(61/4)*x**24*(a/(b*x**4) + 1) **(1/4)*gamma(-21/4)/(256*a**7*b**9*x**20*gamma(-5/4) + 768*a**6*b**10*x** 24*gamma(-5/4) + 768*a**5*b**11*x**28*gamma(-5/4) + 256*a**4*b**12*x**32*g amma(-5/4)) + 1056*a*b**(65/4)*x**28*(a/(b*x**4) + 1)**(1/4)*gamma(-21/...
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {\frac {1547 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} b^{3}}{x^{9}} - \frac {3213 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} b^{2}}{x^{13}} + \frac {2457 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} b}{x^{17}} - \frac {663 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}}}{x^{21}}}{13923 \, a^{4}} \] Input:
integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="maxima")
Output:
1/13923*(1547*(b*x^4 + a)^(9/4)*b^3/x^9 - 3213*(b*x^4 + a)^(13/4)*b^2/x^13 + 2457*(b*x^4 + a)^(17/4)*b/x^17 - 663*(b*x^4 + a)^(21/4)/x^21)/a^4
\[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{22}} \,d x } \] Input:
integrate((b*x^4+a)^(5/4)/x^22,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(5/4)/x^22, x)
Time = 2.00 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {128\,b^5\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^4\,x}-\frac {22\,b\,{\left (b\,x^4+a\right )}^{1/4}}{357\,x^{17}}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{21\,x^{21}}-\frac {32\,b^4\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^3\,x^5}+\frac {20\,b^3\,{\left (b\,x^4+a\right )}^{1/4}}{13923\,a^2\,x^9}-\frac {5\,b^2\,{\left (b\,x^4+a\right )}^{1/4}}{4641\,a\,x^{13}} \] Input:
int((a + b*x^4)^(5/4)/x^22,x)
Output:
(128*b^5*(a + b*x^4)^(1/4))/(13923*a^4*x) - (22*b*(a + b*x^4)^(1/4))/(357* x^17) - (a*(a + b*x^4)^(1/4))/(21*x^21) - (32*b^4*(a + b*x^4)^(1/4))/(1392 3*a^3*x^5) + (20*b^3*(a + b*x^4)^(1/4))/(13923*a^2*x^9) - (5*b^2*(a + b*x^ 4)^(1/4))/(4641*a*x^13)
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^{22}} \, dx=\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (128 b^{5} x^{20}-32 a \,b^{4} x^{16}+20 a^{2} b^{3} x^{12}-15 a^{3} b^{2} x^{8}-858 a^{4} b \,x^{4}-663 a^{5}\right )}{13923 a^{4} x^{21}} \] Input:
int((b*x^4+a)^(5/4)/x^22,x)
Output:
((a + b*x**4)**(1/4)*( - 663*a**5 - 858*a**4*b*x**4 - 15*a**3*b**2*x**8 + 20*a**2*b**3*x**12 - 32*a*b**4*x**16 + 128*b**5*x**20))/(13923*a**4*x**21)